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The Isoperimetric Inequality Via Approximation Theory and Free Boundary Problems

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The Isoperimetric Inequality Via Approximation Theory and Free Boundary Problems THE ISOPERIMETRIC INEQUALITY VIA APPROXIMATION THEORY AND FREE BOUNDARY PROBLEMS CATHERINE BEN¶ ETEAU¶ AND DMITRY KHAVINSON Abstract. In this survey paper, we examine the isoperimetric inequality from an analytic point of view. We use as a point of departure the concept of analytic content in approximation theory: this approach reveals ties to overdetermined boundary problems and hydrodynamics. In particular, we look at problems connected to determining the shape of an electri¯ed droplet or equivalently, that of an air bubble in fluid flow. We also discuss the connection with the Schwarz function and quadrature domains. Finally, we survey some known generalizations to higher dimensions and list many open problems that remain. This paper is an expanded version of the plenary talk given by the second author at the fourth CMFT conference in Joensuu, Finland, in June 2005. 1. History The isoperimetric theorem deals with the question of ¯nding, among all simple closed curves of a given length P , the curve that surrounds the largest area. It was already known to the Greeks that the answer is the circle of radius R = P=2¼ : Pappus and later Theon of Alexandria (III A.D.) already state the result and give credit to Zenodorus (see [4, 6] for a historical discussion). We thus obtain the isoperimetric inequality: (1.1) Area = A · ¼(P=2¼)2 , 4¼A · P 2: \Cutting a hole" in a domain will increase P and diminish A, so (1.1) holds for multiply connected domains as well. J. Steiner [58] in 1838 gave the ¯rst proof of the result assuming the solution exists. It was completed by P. Edler [15] in 1882, H.A. Schwarz [54] in 1884 and C. Carath¶eodory and E. Study [8] in 1910. Schwarz also proved the three dimensional version S3 ¸ 36¼V 2: E. Schmidt [53] in 1938 published a proof of the n-dimensional version: Sn ¸ 2¼n=2nn¡1V n¡1=¡(n=2): A survey by BlÄasjÄo[6], giving in particular a nice account of the early geometric \proofs" of the isoperimetric theorem, has recently appeared. One of the early analytic proofs of (1.1) was given by A. Hurwitz [26] in 1901. Let us sketch his argument here. 2000 Mathematics Subject Classi¯cation. Primary: 30H05, Secondary: 35R35. The second author gratefully acknowledges support from the National Science Foundation. 1 2 C. BEN¶ ETEAU¶ AND D. KHAVINSON Sketch of proof. Suppose the region ­ is bounded by the simple closed smooth curve ¡; parametrized with respect to the arc-length parameter s and with length 2¼: (So, the isoperimetric inequality would state that A · ¼:) That is, we can write X1 ins ¡ := fz(s) = cne g; ¡1 where jz0(s)j = 1: Then Z 1 2¼ X1 1 = jz0(s)j2ds = n2jc j2: 2¼ n 0 ¡1 On the other hand, using Green's theorem, Z 1 A = (xdy ¡ ydx) 2 ¡ Z 1 = Im z¹(s)z0(s)ds 2 ¡ X1 2 = ¼ njcnj : ¡1 2 Since n · n , A · ¼. A = ¼ , cn = 0; n 6= 0; 1, i.e. ins ¡ := fc0 + c1e g; a circle: 2 Another proof using the Riemann mapping theorem was given by Carleman in 1921. Theorem 1.1. ([9]) Suppose ­ is a Jordan domain with a recti¯able boundary. Then for all f analytic in ­ and continuous in the closure of ­; ZZ Z 1 jfj2dA · ( jfjds)2: ­ 4¼ ¡ Notice that letting f = 1 yields the isoperimetric inequality. Also noteR that the 1 2 above inequality would be rather trivial if we had the expression 4¼ ¡ jfj ds on the right hand side; that inequality can be obtained immediately from the similar inequality in the unit disk via conformal mapping. The Carleman inequality, however, requires an additional neat trick beyond a change of variables. In this survey paper, we are interested in the analytic approach to the isoperimet- ric inequality and its connections with approximation theory and free boundary problems. This approach reveals a close tie to hydrodynamics and, in particu- lar, to problems concerning shapes of electri¯ed droplets of perfectly conducting fluid. We use as a point of departure the paper [31], in which the author discusses THE ISOPERIMETRIC INEQUALITY 3 the concept of analytic content, and the related survey paper [18]. In Section 3, we discuss the connection with overdetermined boundary value problems and Serrin's theorem. In Section 4, we describe a more general problem and its appli- cation to determining the shape of a droplet of conducting fluid in the presence of an electric ¯eld. In Section 5, we examine some special cases of this more general problem in connection with the Schwarz function, Vekua's problem, and quadrature domains. Finally, in Section 6, we survey known generalizations to higher dimensions and state many remaining open problems. Acknowledgements. The authors would like to thank Dr. Van Deursen and Dr. Darren Crowdy for pointing out several interesting references to electrowetting. 2. Analytic Content Let ­ be a ¯nitely connected region in C bounded by n simple closed analytic curves γj; j = 1; : : : ; n: Let ¡ = [γj be the boundary of the region ­:A­ denotes the set of functions analytic in ­ and continuous in ­¹; and C(­)¹ is the set of continuous functions in ­¹: Let R(­)¹ denote the closure (in the uniform norm) of the set of rational functions with poles outside of ­¹: In general, not every function analytic in ­ and continuous in ­¹ is an element of R(­)¹ ; but it is well- known that for the ¯nitely connected regions ­ with analytic boundaries that ¹ we are considering here, A­ = R(­) (see, for example, [16]). In general, it is an interesting and central question of approximation theory to determine when a function is an element of R(­)¹ ; or what its distance to R(­)¹ is. In particular, we will be interested in approximating the \simplest" non-analytic function, namely z:¹ The following de¯nition was introduced in [30]. De¯nition. The analytic content of a domain ­ is ¸(­) := inf k z¹ ¡ ' kC(­)¹ : '2A­ Therefore the analytic content of a compact set ­¹ may be thought of as the distance between the functionz ¹ and the algebra R(­)¹ ; or in the case of ¯nitely connected domains, A­: The following theorem gives a quantitative geometric estimate of the analytic content of a set - and also yet another proof of the isoperimetric inequality! Theorem 2.1. Let ­ and ¡ be as above, and let A and P be the area and perimeter of ­; respectively. Then r 2A A · ¸(­) · ; P ¼ q 2 A so P ¸ 4¼A: Moreover, ¸(­) = ¼ , ­ = disk: This theorem is discussed in detail in [18]. Let us outline the argument here. H. Alexander in 1973 ([3]) proved the upper estimate by noticing the connection with the Ahlfors-Beurling estimate from 1950 ([1]). 4 C. BEN¶ ETEAU¶ AND D. KHAVINSON More speci¯cally, suppose D is a bounded domain (with smooth boundary @D) containing ­¹: By the Cauchy-Green formula, Z Z 1 z¹ 1 1 ³¹ = dz ¡ dA(z); 2¼i @D z ¡ ³ ¼ D z ¡ ³ where dA is area measure. De¯ne Z 1 1 G(³) = dA(z): ¼ ­¹ z ¡ ³ Then Z Z 1 z¹ 1 1 ³¹ + G(³) = dz ¡ dA(z): 2¼i @D z ¡ ³ ¼ D¡­¹ z ¡ ³ The right hand side can be easily seen to be in A­: Therefore ¸ · max jG(³)j: ³2­¹ The Ahlfors-Beurling estimate shows that r A jG(³)j · ; ¼ with equality holding only for a disk, thus proving the theorem. Gamelin and D. Khavinson gave a simple proof of the Ahlfors-Beurling estimate (see [18, p. 25- 28]). Their argument goes as follows. Without loss of generality, max³2­¹ jG(³)j is attained at z = 0, and, rotating, equals Z 1 dA(³) Re = ¸max: ¼ ­ ³ Now notice that f³ : Re 1 ¸ cg = f³ : j³ ¡ 1 j · 1 g =: ¢, a disk. For this disk, q ³ 2c 2c 1 A(¢) ¸ = 2c = ¼ ; while it is easy to show that, for any ­ 6= ¢; the integral is smaller, as desired. D. Khavinson proved the lower estimate in [30] by applying the complex form of Green's theorem to the functionz ¹¡'(z); where ' 2 A­ is the best approximation toz: ¹ This gives Z Z (¹z ¡ '(z))dz = 2i dA(z) = 2iA; ¡ ­ so that Z ¸P ¸ j (¹z ¡ '(z))dzj ¸ 2A: ¡ Theorem 2.1 leads naturally to the following: 2A Question. For which ­ does the equality P = ¸(­) hold? The next theorem and following remarks are taken from [31]. THE ISOPERIMETRIC INEQUALITY 5 Theorem 2.2. ([31]) Let ­ and ¡ be as above. The following are equivalent: 2A (i) ¸ = P ; (ii)There is ' 2 A­ such that z¹(s) ¡ i¸z¹_(s) = '(z(s)) on ¡; where s is the arc-length parameter; R R 1 1 (iii) A ­ fdA = P ¡ fds for all f 2 A­: Remark. Note that (iii) holds for annuli ­ = fr < jzj < Rg. Simply take the Laurent series decomposition of f = f1 + f2 in the annulus, where f1 is analytic inside fz : jzj < Rg and f2(1) = 0; and notice that both sides of the equality in (iii) are equal to f1(0): Also, (ii) easily implies that if ¡ contains a circular arc, ­ is a disk or an annulus. Indeed, suppose, for simplicity, that ¡ contains an arc centered at the origin and of radius R: Then we can write z(s) = Reis=R; where s is the arc-length parameter, and a calculation shows that R2 ¡ R¸ c z¹(s) ¡ i¸z¹_(s) = = = '(z) z(s) z on that arc. But since ' 2 A­; this equality must hold on all of ¡: If c = 0; then ' = 0 and jzj = const on ¡; so ¡ is a disk centered at the origin. If c 6= 0; i¸zz¹_(s) = jzj2 ¡ c; ¹ 1 djzj2 2 or, in other words, Re(zz_(s)) = 2 ds = 0: Therefore jzj is a constant locally on @­; so ­ is an annulus.
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